Translation and 2D Plane Groups

II. Lecture 2: Translation and 2D Plane Groups Crystals are three-dimensional periodic arrays of a group of atoms. This group of atoms may be (SiO4)-4 or H2O, for example, and is called a motif. The motif is repeated via vector translations in three dimensions and forms a lattice. In earth materials, the translations are on the order of 0.5 to 2.0 nm. Translations and Lattices in 2D In diagram below, the motif is represented by points called lattice points. Note that the environment of every point is identical to that of every other point. The polygon building blocks that compose a lattice are called unit cells. Primitive unit cells contain only lattice nodes and multiple unit cells contain an extra node within the cell. The smallest unit cell is called the reduced unit cell In two dimensions, there are only five distinct lattices. Row 1 b Origin a �= y 90o b Origin a � = 60o (A) b Origin a � (C) a' b' x x Row 1 Row 1 (D) y y Row 1 x b Origin a � = 90o y b Origin a � (E) x y' x Row 1 y (B) b' a x (Adapted from Klein and Hurlbut, Jr. Manual of Mineralogy. New York: John Wiley and Sons, Inc., 1993. 113) Combining Point Symmetry Operations with 2D Translations A lattice must be compatible with the point symmetry of the motif it repeats. This restraint limits the motif’s possible symmetries. Understanding how requires a mathematical approach to symmetry operations. Vector and Matrix Digression Performing inversion or reflection operations requires matrix multiplication. Example 1. Invert point (x, y, z) through the origin to find (x’, y’, z’). I represents the inversion matrix. Example 2. Develop the generic formula for the rotation of (x, y, z) about the axis defined by x = 0, y = 0. Back to Rotations and Translations For a lattice point to have rotational symmetry, the points at the end of the rotated unit vector r must also be lattice points. r = cos (2π/n) i + sin (2π/n) j Overlaying point operations like rotation, reflection, and glide reflection on the five distinct plane lattices yields 17 plane groups. These plane groups express all the symmetry combinations possible in a plane. p1 pm p2 p2mm c2mm p4 pg cm p2gg p2mg p3 p3m1 p4gm p4mm p31m p6 p6mm (Adapted from Klein and Hurlbut, Jr. Manual of Mineralogy. New York: John Wiley and Sons, Inc., 1993. )